PbTiO3 (perovskite)

PbTiO₃ has a simple perovskite structure. Many compounds of this type undergo one or more structural phase transitions. PbTiO₃ undergoes a cubic-to-tetragonal ferroelectric phase transition at around 770K.
In terms of the strain in a thin film, equilibrium theories of epitaxy predict that, below a critical thickness, the strain coming from the lattice mismatch will be accommodated by film itself. Above the thickness, the strain will be partially relaxed by forming dislocations. If the film undergoes a strutural phase transition from a high-symmetry phase to a low-symmetry phase during cooling from the growth temperature, the epitaxial strain can be relieved by domain formation as suggested by Roitburd and Bruinsma and Zangwill.
The potential surface was initially mapped out using the Linear Augmented PlaneWave (LAPW) method, and the charge density and electronic struccure were analyzed. PbTiO₃ showed a much deeper well when tetragonal strain was included. Thus the tetragonal strain is responsible for the tetragonal ground state in PhTiO3. In BaTiO3, the Ba is quite spherical in the ferroelectric phase, whereas the Pb in PhTiO3 is not very spherical in the ferroelectric phase, and polarization of the Pb helps stabilize the large strain and the tetragonal ground state in PbTiO3.
In PbTiO3, the O 2p states strongly hybridize with the d₀ Ti⁴⁺ cation, which reduces the short-range repulsions thus allows off-center displacements.

Elastic constants of PbTiO₃:
c₁₁=1.433
c₁₂=0.322
c₁₃=0.241
c₃₃=1.316
c₄₄=0.558
c₆₆=0.556
Y=1.34
μ=0.58
ν=0.16
All terms except ν have been divided by a factor of 10¹¹N/m²

Some constants of MgO:
Y=3.105
μ=1.332
ν=0.161
Ref:B.S.Kwak, Physical Review B, Vol49, 14865, 1994

Lattice constants(Å) and TEC of various materials:
PbTiO₃:
tetragonal(RT), a=3.899(bulk); c=4.153(bulk);
cubic (823K), a=3.986,
TEC=12.6 x 10^-6.
Room temperature crystal structure of tetragonal PbTiO₃ was determined by Shirane et al.(1956) with displacements parallel to the polar axis (relative to the Pb ion at the origin):
dz_Ti=17pm
dz_OI=dz_OII=47pm

Note: Oxygen octahedra suffers no distortion in going to ferroelectric phase.
KTaO₃:
cubic(RT),a=3.989;
cubic(823K),a=4.003;
TEC=6.67 x 10^-6.
STO:
cubic(RT),a=3.905;
cubic(823K),a=3.928;
TEC=11.7 x 10^-6.
ferroelectric phase transition: 35-40K
MgO (NaCl structure):
cubic(RT),a=4.213;
cubic(823K),a=4.239;
TEC=14.8 x 10^-6.
BaTiO₃
In the tetragonal phase, the Ti and O ions move relative to Ba at the origin from their cubic position:
Ti: (1/2,1/2,1/2) to (1/2,1/2,1/2+dz_Ti)
O: (1/2,1/2,0), (1/2,0,1/2) and (0,1/2,1/2) to (1/2,1/2,dz_OI),(1/2,0,1/2+dz_OII) and (0,1/2,1/2-dz_OII
dz_Ti=5pm
dz_OI=-9pm
dz_OII=-6pm
(Harada et al.1970)
prototype cubic perovskite: >120°C;
ferro 4mm tetragonal: 120°C>T>5°C;
ferro mm orthorhombic: 5°C>T>-90°C;
ferro 3m trigonal: <-90°C
PbZrO₃:
Antiferroelectricity
Paraelectric-antiferroelectric: 230°C in zero field
but upon application of an external electric field below T_c will induce a transition to a rhombohedral ferroelectric phase.

Ref: Principles and Applications of Ferroelectrics and Related Materials by M.E. Lines and A.M. Glass

Developments of Multiferroic

Multiferroic magnetoelectrics are materials that are both ferromagnetic and ferroelectric in the same phase. For example, nickel boracite, Ni₃B₇O₁₃I. However, very few exist naturally or have been synthesized in the lab. It was proposed by N.A. Hill that the transition metal d electrons, which are essential for magnetism, reduce the tendency for off-center ferroelectric distortion.
Considering the origin of ferromagnetism, it is useful to treat the interaction as shifting the energy of the 3d band for electrons with one spin direction relative to the band for electrons with the opposite spin direction. In the stoner theory, the fundamental driving force for ferromagnetism is the exchange energy, which is minimized if all of the electrons have the same spin. However, transfering electrons from the lowest band states (occupied equally with up and down electrons) to band states of higher energy will increase the band energy. And this band energy prevents simple metals from being ferromagnetic. Depending on the 3d and 4s band structures and the Fermi energy level, some metals are ferromagnetic, such as, Fe(3d⁶4s²), Co(3d⁷4s²) and Ni(3d⁸4s²), some are not ferromagnetic, like Cu(3d¹⁰4s¹) and Zn(3d¹⁰4s²).
For ferroelectricity, it is a matter of two competing forces. One is the short-range repulsion between adjacent electron clouds, which favor the nonferroelectric symmetric structure. The opposing force is the additional bonding considerations, which might stabilize the ferroelectric phase. The short-range repulsions dominate at high temperature, resulting in the symmetric, unpolarized state. As the temperature is decreased, the stabilizing force become stronger and the polarized state becomes stable.
In 1961, G.T.Rado and VJ Folen first observed the magnetoelectric effect in Cr₂O₃

Terminologies related to Multiferroics

Ferromagnets:
A ferromagnetic material is one that undergoes a phase transition from a high-temperature phase that does not have a macroscopic magnetic moment to a low-temperature phase that has a spontaneous magnetization even in the absence of an
applied magnetic field.

ferromanetic:
--> -->
--> -->

ferrimagnetic:

-> ->
<-- <--  

antiferromagnetic:
--> -->
<-- <-- 

Ferroelectric:
Ferroelectric materials is one that undergoes phase transition from a high-temperature phase that behaves as an ordinary dielectric (so that an applied electric filed induces an electric polarization, which goes to zero when the field is removed) to a low-termperature phase with a spontaneous electric polarization whose direction can be switched by an applied field.

d₃₃, charged constant:
One can use either the direct piezoelectric effect, i.e., applying a stress and measuring the induced charge (or voltage), or use the inverse piezoelectric effect, i.e., applying a voltage (field) and measuring the induced strain.
d₃₃=(∂S₃/∂E₃)_T
Or
d₃₃=(∂D₃/∂T₃)_E
S:strain
E:electric field
D: electric displacement
T: stress
x₃ axis is the poling direction. For thin film, usually it is the normal direction to the surface.

Tensors:
piezoelectric constatnt: third-rank tensor
electrostrictive constant and elastic constant: fourth-rank tensors.
Tensor notation/matrix notation: 11/1, 22/2, 33/3, 23(32)/4, 31(13)/5, 12(21)/6

Three dimensional heteroepitaxy in self-assembled nanostructures

Take BaTiO₃-CoFe₂O₄ as an example:
1. Intrinsic similarity in crystal chemistry between perovskite and spinel. Both of them have octahedral oxygen coordination. Lattice mismatch is about 5%. CoFe₂O₄: a=0.838nm
2. Very little solid solubility in each other
3. Suitable substrate with similiar crystal structure, such as STO